Douglas Blumeyer's RTT How-To: Difference between revisions

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Cents and Hertz values can readily be converted between one form and the other, so it’s the second difference which is more important. It’s their size. If we do convert 12-ET’s generator to cents so we can compare it with meantone’s generator at 12-ET, we can see that 12-ET’s generator is 100¢ (log₂1.059 × 1200¢ = 100¢) while meantone’s generator at 12-ET is 500¢ (5/12 × 1200¢ = 500¢). What is the explanation for this difference?

Well, notice that meantone is not the only temperament which passes through 12-ET. Consider augmented temperament. Its generator at 12-ET is 400¢. What's key here is that all three of these generators — 100¢, 500¢, and 400¢ — are multiples of 100¢.

Well, notice that meantone is not the only temperament which passes through 12-ET. Consider augmented temperament. Its generator at 12-ET is 400¢<ref>This statement is slightly misleading, in order to help make the more important point it's in the context of. The full truth is that augmented's ''period'' is 400¢ at 12-ET. As you'll soon seen, the period is just a special name for the first one of a temperament's generators, but because in rank-2 situations like this where there are only two generators, referring to ''the'' generator typically implies the second generator, the one which is not the period. For augmented, this generator is 100¢ at 12-ET.</ref>. What's key here is that all three of these generators — 100¢, 500¢, and 400¢ — are multiples of 100¢.

Let’s put it this way. When we look at 12-ET in terms of itself, rather than in terms of any particular rank-2 temperament, its generator is 1\12. That’s the simplest, smallest generator which if we iterate it 12 times will touch every pitch in 12-ET. But when we look at 12-ET not as the end goal, but rather as a foundation upon which we could work with a given temperament, things change. We don’t necessarily need to include every pitch in 12-ET to realize a temperament it supports.

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Here’s a diagram that shows how the generator size changes gradually across each line in PTS. It may seem weird how the same generator size appears in multiple different places across the space. But keep in mind that pretty much any generator is possible pretty much anywhere here. This is simply the generator size pattern you get when you lock the period to exactly 1200 cents, to establish a common basis for comparison. These are called [[Tour_of_Regular_Temperaments#Rank-2_temperaments|linear temperaments]]. This is what enables us to produce maps of temperaments such as the one found at [[Map_of_linear_temperaments|this Xen wiki page]], or this chart here ''(see Figure 4b)''.

[[File:Generator sizes in PTS.png|800px|thumb|'''Figure 4b.''' Generator sizes of linear temperaments in PTS. Don't worry too much about the valid ranges yet; we'll discuss that part later. ~~The temperament lines that aren't labelled in this diagram have non-octave periods; they are rank-2, but not linear, and it doesn't make enough sense to compare their generators here.]]~~

[[File:Generator sizes in PTS.png|800px|thumb|'''Figure 4b.''' Generator sizes of linear temperaments in PTS. Don't worry too much about the valid ranges yet; we'll discuss that part later. And I didn’t break down what’s happening along the blackwood, compton, augmented, dimipent, and some other lines which are labelled on the original PTS diagram. In some cases, it’s just because I got lazy and didn’t want to deal with fitting more numbers on this thing. But in the case of all those that I just listed, it’s because those temperaments all have non-octave periods; they are rank-2, but therefore not linear, and so it doesn't make enough sense to compare their generators here. You'll learn about periods in the next section.]]

And ~~note that~~ I didn’t break down what’s happening along the blackwood, compton, augmented, dimipent, and some other lines which are labelled on the original PTS diagram. In some cases, it’s just because I got lazy and didn’t want to deal with fitting more numbers on this thing. But in the case of all those that I just listed, it’s because those temperaments all have non-octave periods.

Let’s bring up MOS theory again. We mentioned earlier that you might have been familiar with the scale tree if you’d worked with MOS scales before, and if so, the connection was scale cardinalities, or in other words, how many notes are ~~in the resultant scales when you continuously iterate the generator until you reach points where there are only two scale step sizes. At these points scales tend to sound relatively good~~, ~~and this is in fact the definition of a MOS scale. There’s a mathematical explanation for how to know~~, ~~given a ratio between the size of your generator~~ and ~~period, the cardinalities of scales possible; we won’t re-explain it here. The point is that the scale tree can show you that pattern visually. And~~ so ~~if each temperament line in PTS is its own segment of the scale tree, then we can use~~ it ~~in a similar way.~~

For example, if we pick a point along the meantone line between 46 and 29, the cardinalities will be 5, 12, 17, 29, 46, etc. If we chose exactly the point at 29, then the cardinality pattern would terminate there, or in other words, eventually we’ll hit a scale with 29 notes and instead of two different step sizes there would only be one, and there’s no place else to ~~go from there~~. The system has circled back around to its starting point, so it’s a closed system. Further generator iterations will only retread notes you’ve already touched. The same would be true if you chose exactly the point at 46, except that’s where you’d hit an ET instead.

~~Between ETs,~~ in the stretches of rank-2 temperament lines where the generator is not a rational fraction of the octave, theoretically those temperaments could have infinite pitches; you could continuously iterate the generator and you’d never exactly circle back to the point where you started. If bigger numbers were shown on PTS, you could continue to use those numbers to guide your cardinalities forever.

~~The structure when you stop iterating the meantone generator with five notes is called meantone[5~~]~~. If you were to use the entirety of 12-ET as meantone then that’d be meantone[12~~]. But you can also realize meantone[12] in 19-ET; in the former you have only one step size, but in the latter you have two. You can’t realize meantone[19] in 12-ET, but you could also realize it in 31-ET.

=== periods and generators ===

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 Cents and Hertz values can readily be converted between one form and the other, so it’s the second difference which is more important. It’s their size. If we do convert 12-ET’s generator to cents so we can compare it with meantone’s generator at 12-ET, we can see that 12-ET’s generator is 100¢ (log₂1.059 × 1200¢ = 100¢) while meantone’s generator at 12-ET is 500¢ (5/12 × 1200¢ = 500¢). What is the explanation for this difference?
-Well, notice that meantone is not the only temperament which passes through 12-ET. Consider augmented temperament. Its generator at 12-ET is 400¢. What's key here is that all three of these generators — 100¢, 500¢, and 400¢ — are multiples of 100¢.
+Well, notice that meantone is not the only temperament which passes through 12-ET. Consider augmented temperament. Its generator at 12-ET is 400¢<ref>This statement is slightly misleading, in order to help make the more important point it's in the context of. The full truth is that augmented's ''period'' is 400¢ at 12-ET. As you'll soon seen, the period is just a special name for the first one of a temperament's generators, but because in rank-2 situations like this where there are only two generators, referring to ''the'' generator typically implies the second generator, the one which is not the period. For augmented, this generator is 100¢ at 12-ET.</ref>. What's key here is that all three of these generators — 100¢, 500¢, and 400¢ — are multiples of 100¢.
 Let’s put it this way. When we look at 12-ET in terms of itself, rather than in terms of any particular rank-2 temperament, its generator is 1\12. That’s the simplest, smallest generator which if we iterate it 12 times will touch every pitch in 12-ET. But when we look at 12-ET not as the end goal, but rather as a foundation upon which we could work with a given temperament, things change. We don’t necessarily need to include every pitch in 12-ET to realize a temperament it supports.
@@ Line 398: / Line 398: @@
 Here’s a diagram that shows how the generator size changes gradually across each line in PTS. It may seem weird how the same generator size appears in multiple different places across the space. But keep in mind that pretty much any generator is possible pretty much anywhere here. This is simply the generator size pattern you get when you lock the period to exactly 1200 cents, to establish a common basis for comparison. These are called [[Tour_of_Regular_Temperaments#Rank-2_temperaments|linear temperaments]]. This is what enables us to produce maps of temperaments such as the one found at [[Map_of_linear_temperaments|this Xen wiki page]], or this chart here ''(see Figure 4b)''.
-[[File:Generator sizes in PTS.png|800px|thumb|'''Figure 4b.''' Generator sizes of linear temperaments in PTS. Don't worry too much about the valid ranges yet; we'll discuss that part later. The temperament lines that aren't labelled in this diagram have non-octave periods; they are rank-2, but not linear, and it doesn't make enough sense to compare their generators here.]]
+[[File:Generator sizes in PTS.png|800px|thumb|'''Figure 4b.''' Generator sizes of linear temperaments in PTS. Don't worry too much about the valid ranges yet; we'll discuss that part later. And I didn’t break down what’s happening along the blackwood, compton, augmented, dimipent, and some other lines which are labelled on the original PTS diagram. In some cases, it’s just because I got lazy and didn’t want to deal with fitting more numbers on this thing. But in the case of all those that I just listed, it’s because those temperaments all have non-octave periods; they are rank-2, but therefore not linear, and so it doesn't make enough sense to compare their generators here. You'll learn about periods in the next section.]]
-And note that I didn’t break down what’s happening along the blackwood, compton, augmented, dimipent, and some other lines which are labelled on the original PTS diagram. In some cases, it’s just because I got lazy and didn’t want to deal with fitting more numbers on this thing. But in the case of all those that I just listed, it’s because those temperaments all have non-octave periods.
-Let’s bring up MOS theory again. We mentioned earlier that you might have been familiar with the scale tree if you’d worked with MOS scales before, and if so, the connection was scale cardinalities, or in other words, how many notes are in the resultant scales when you continuously iterate the generator until you reach points where there are only two scale step sizes. At these points scales tend to sound relatively good, and this is in fact the definition of a MOS scale. There’s a mathematical explanation for how to know, given a ratio between the size of your generator and period, the cardinalities of scales possible; we won’t re-explain it here. The point is that the scale tree can show you that pattern visually. And so if each temperament line in PTS is its own segment of the scale tree, then we can use it in a similar way.
-For example, if we pick a point along the meantone line between 46 and 29, the cardinalities will be 5, 12, 17, 29, 46, etc. If we chose exactly the point at 29, then the cardinality pattern would terminate there, or in other words, eventually we’ll hit a scale with 29 notes and instead of two different step sizes there would only be one, and there’s no place else to go from there. The system has circled back around to its starting point, so it’s a closed system. Further generator iterations will only retread notes you’ve already touched. The same would be true if you chose exactly the point at 46, except that’s where you’d hit an ET instead.
-Between ETs, in the stretches of rank-2 temperament lines where the generator is not a rational fraction of the octave, theoretically those temperaments could have infinite pitches; you could continuously iterate the generator and you’d never exactly circle back to the point where you started. If bigger numbers were shown on PTS, you could continue to use those numbers to guide your cardinalities forever.
-The structure when you stop iterating the meantone generator with five notes is called meantone[5]. If you were to use the entirety of 12-ET as meantone then that’d be meantone[12]. But you can also realize meantone[12] in 19-ET; in the former you have only one step size, but in the latter you have two. You can’t realize meantone[19] in 12-ET, but you could also realize it in 31-ET.
 === periods and generators ===